Editing Euclidean space (section)

===Technical definition===

A '''{{vanchor|Euclidean vector space}}''' is a finite-dimensional [[inner product space]] over the [[real number]]s.{{sfn|Berger|1987|loc=Chapter 9}}

A '''Euclidean space''' is an [[affine space]] over the [[real number|reals]] such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called ''Euclidean affine spaces'' to distinguish them from Euclidean vector spaces.{{sfn|Berger|1987|loc=Chapter 9}}

If {{mvar|E}} is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted <math>\overrightarrow E.</math> The ''dimension'' of a Euclidean space is the [[dimension (vector space)|dimension]] of its associated vector space.

The elements of {{mvar|E}} are called ''points'', and are commonly denoted by capital letters. The elements of <math>\overrightarrow E</math> are called ''[[Euclidean vector]]s'' or ''[[free vector]]s''. They are also called ''translations'', although, properly speaking, a [[translation (geometry)|translation]] is the [[geometric transformation]] resulting from the [[group action|action]] of a Euclidean vector on the Euclidean space.

The action of a translation {{mvar|v}} on a point {{mvar|P}} provides a point that is denoted {{math|''P'' + ''v''}}. This action satisfies

<math display="block">P+(v+w)= (P+v)+w.</math>

'''Note:''' The second {{math|+}} in the left-hand side is a vector addition; each other {{math|+}} denotes an action of a vector on a point. This notation is not ambiguous, as, to distinguish between the two meanings of {{math|+}}, it suffices to look at the nature of its left argument.

The fact that the action is free and transitive means that, for every pair of points {{math|(''P'', ''Q'')}}, there is exactly one [[displacement (geometry)|displacement vector]] {{mvar|v}} such that {{math|1=''P'' + ''v'' = ''Q''}}. This vector {{mvar|v}} is denoted {{math|''Q'' − ''P''}} or <math>\overrightarrow {PQ}\vphantom{\frac){}}.</math>

As previously explained, some of the basic properties of Euclidean spaces result from the structure of affine space. They are described in {{slink||Affine structure}} and its subsections. The properties resulting from the inner product are explained in {{slink||Metric structure}} and its subsections.